4.94 
FLUXIONS, 
Ex. 3. Let AC be the involute of a circle ; to find its 
length. Here t is conftant, by Art. 51. Ex. 4. hence, 
y 2 £2 
C; but when 2—0, yzzt, .*. o~— 4*C, and 
~ t 2 , jr 2 —t 2 SY 2 
Gzz-: hence, 2='-—-. 
2 1 21 2SA 
To FIND THE SURFACES OF SOLIDS. 
Prop. XXIV. To find, the J'urface of a fiolid generated ly 
the rotation of a curve about its axis, or by the motion of a plane 
parallel to itfielfi.. 
55. Conceiving the folid AFH to be generated as in 
Art. 52, by the circle CD, the furface may be confidered 
E as generated by the periphery 
of the circle; the fluxion there¬ 
fore of the furface will be the 
periphery of the circle multi¬ 
plied by the velocity with 
which it flows, by Cor. Art.49. 
But the velocity with which 
any point C of the periphery 
flows, is the velocity with 
which AC increafes at the 
point C, or it is z, putting 
AC=z, Hence, if we put 
Tl AB— x, BCzziy, p—6,1 831S, 
8c c. the circumference of a circle whofe radius —1 (Art. 
53. Ex. 4.) S z= the furface ACD ; then 1 :y:: p : py the 
circumference of the circle CD ; therefore S zzpyz the 
fluxion of the furface; confequently the fluent of pyz, 
corrected if r.eceffary, will be the furface. 
This method of finding the fluxion of the furface of 
a folid, may be further illuftrated thus. Let ACF be 
protended into a ftraight line, and let an ordinate perpen¬ 
dicular to it, and always equal to the periphery of the 
circle CD, move from A to F with the fame velocity as 
the point C upon the folid moves; then it is manifeft, 
that the area generated by this ordinate muff always be 
equal to the area generated by the periphery of the cir¬ 
cle, the generating lines and their velocities being always 
equal, and both moving in directions perpendicular to 
themfelves ; hence, the fluxion of the furface ACD 22 
the fluxion of the area of this curve — (by Art. 49.) the 
ordinate multiplied by the fluxion of the abfcilfa — the 
periphery of the circle CD multiplied by the fluxion of 
the curve AC. 
Ex. 1. Let ADFC be a fphere whofe center is O ; to 
find its furface. Let Cs be a tangent at C, sEw parallel 
to BC, and CE to B m ; then if 
AB=x-, BC—9>, ACxz, by Art. 
23. CEzzzv; and by fimi- 
lar triangles CEs-, CBO, z : x :: 
a : y, . •. yz—ax ; hence, Sz -pyz, 
— pax, the fluxion of the furface 
DAC, whofe fluent Szzpax+C ; 
but when x—o, S—o, .•. C=;o; 
hence, S —pax the furface DAC. 
If we make AB equal to AE, or 
xzzzaye have 2paP for the whole 
furface of the fphere. Now if 
we conceive ADFC to be a great 
circle of the fphere, its area = ipa 2 , by Art. 49. Ex. 2. 
A. 
*171 
/IV 
/ E \ 
0 
Cor. Hence, the whole furface of a fphere is equal to 
four times the area of a great circle of that fphere. 
Cor. As the furface DAC —pax, it varies as x. 
Ex. 2. Let the fol*d AFH be generated by the common 
parabola ; to find its furface. Here axzzy 2 ; hence, x— 
hence, S —pyz— ---whofe fluent, by Art. 
a 
39- 
is S—C ; now when^'zro, S=o, in 
I 2(2 
which cafe, the equation becomes o~— + C j hence, 
12 
pa- 
therefore S= ^f±£!l 
12 12a 12 
Ex. 3. Let ALN be a groin, as in Art. 52. Ex. 7. to 
find its furface. Rut ABzzzx, BC y, ACzzrz; and we 
have (Article46.) zz 
- ; alfo, vw — 2BC — 
V 2 ax—x* 
2V 2 ax —* 2 ; now vw is the line generating one of the four 
furfaces ; hence, 8 2 ax — x 2 anfwers to py in tlte other 
cafes ; therefore if S be the furface Avx, S = 8 ax, and 
S— Sax-yC ; but when xzzzo, S—o, .‘.C—o; confequently 
Sz=8s;v; and when x—a, S=z8a 2 . 
Ex. 4. To find the furface generated by the revolu¬ 
tion of the cycloidal curve BA about its bafe DA. Put 
By—z, B? — x, rXD—yC—y, BD=«; then, by Art. 53, 
Ex. 2. zzzafx 2 x\ .•. B 
• . i \ 
S zz: pyz — pya 2 x 2 x 
zzpyia—x X a ^ x *XT 
3. _1 i_ i_ 
zz pa 2 x >2 x — pa 2 x 2 x‘, 
i t_ 
lienee, S zz 2 pa 2 x 2 — 
i 3 . 
-Jz2 2 x 2 -|-C ; but when 
x—o, S—o, .•. C—o ; L) 
.2 i 4 J. 
hence, S zz: -ipu 2 x 2 —§ pa 2 x 2 , the furface generated by By, 
and when x=a, we have Szz.^—, the whole furface ge- 
3 
nerated by BA. 
Ex. 5. To find the furface of the folid generated by 
any part CD of the logarithmic curve revolving about 
. jV M 2 + r 2 
its axis AB. By Prop. 22. Ex. 4. zzz--—, there¬ 
fore SzzpyzzzpyV' M 2 fiy 2 , which fluxion is the fame 
as that for the value of z in Prop. 22. Ex. 3. (the 
conflant multiplier and divifor excepted ; therefore Szz: 
L P / 
f -x V 
X V_y 4 _i_M 2 _y 2 
pM* 
X h. 1 . jr+v M 2 +y 2 -f C ; but 
whenjyzz:«, Szz:0; hence, ozz/x l/a 4 -\-M 2 a 2 4- 
pM ! 
X 
h.l.«-l-v/M 2 +a 2 T C, and Czz-^x i/a 4 +M 2 a 2 — 
2 
■ — Xh. 1 . «+^M 2 -p« 2 ; therefore Szz:- X \fy 4 -\-PA 2 y 2 
_^ x viMW+^xh.1.®^. 
2 2 a +VM 2 -\-a 2 
Ex. 6. To find the furface of the folid generated by 
the catenary curve revolving about its axis. Let us 
aflume z 2 zz zax-fx 2 ; hence, a 2 2ax + x 2 zza 2 -fi z 2 , 
z, and j 
and a-\-x~Va 2 —z 2 ; therefore .vzz 
az 
3/z 2 — x 2 : 
3/ a 2 +z 2 
Now Szzpyi,) alfume Szz pyz — w, 
3 yy_ 
a 
and x 2 zz— —(Prop. 22.) z 2 
a 2 
! +i 2 
fyy- 
V a 2 -\-z 2 
then Sz zpyz-\-pzy—w, and as S—pyz, we have wzzpzyzz. 
■fi = 4y2 1 i xi 2 zz . ' iy7Jra2 XJ’ 2 and whofe fluent is w=zpa\/a 2 +z 2 (Prop. 15.) 
^ a 2 J a 2 * " a <\/a 2 -\-z 2 
hence^ 
